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Re: prime generating quadratic conjecture

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  • Patrick Capelle
    ... Thank you, Jens, for this contribution. I would like to give here some comments : 1. There is effectively an error in this mathworld s page. I am going to
    Message 1 of 14 , Apr 1, 2006
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      --- In primenumbers@yahoogroups.com, Wed Mar 29, 2006, "Jens Kruse
      Andersen" <jens.k.a@...> wrote:
      > You may have misunderstood this "limit".
      > If c = 2, 3, 5, 11, 17, or 41, then x^2-x+c is prime for x = 0..c-1.
      > It has been proved there are no other such c values.
      > See http://mathworld.wolfram.com/LuckyNumberofEuler.html
      > (That page incorrectly says x = 0..c-2 but it makes no
      > difference for the solutions).
      > However, there may be large c for which x^2-x+c is prime
      > for x = 0..41 or longer. Finding such a c is infeasible.
      --------------------------------------------------------------------

      Thank you, Jens, for this contribution.
      I would like to give here some comments :

      1. There is effectively an error in this mathworld's page.
      I am going to write a short e-mail to the team of Eric W. Weisstein.
      The polynomials x^2 - x + c are often sources of confusion, and it
      could be also the case here about this definition of the lucky numbers
      of Euler. We don't know if the author wanted to write x = 0 ... c-1
      (because he wanted all the primes, repeated or not) or x = 1 ... c-1
      (because he wanted only the different prime numbers and didn't care to
      begin with x = 1) or x = 0 ... c-2 (because he wanted the same range
      as for the Euler-like polynomials x^2 + x + c).
      But there is another possibility, more plausible : this page correctly
      says that " n = 0 ... p-2 ", but the author put wrongly n^2 - n + p
      instead of n^2 + n + p, whereas he put this last Euler-like
      polynomial in the following link :
      http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
      Note also the following prime curio given by Le Lionnais for the
      number 41 (see http://primes.utm.edu/curios/page.php ) : it's one of
      the numbers p (or "Lucky Numbers of Euler") such that a Euler-like
      polynomial is prime for n = 0, 1, ..., p-2.
      Anyway, the lucky numbers of Euler are concerning the two families of
      polynomials, even if the reference to the mathematician Euler in the
      definition should privilege the Euler-like polynomials n^2 + n + p.
      Do they have to be mentioned both in the definition of the lucky
      numbers of Euler ?

      2. Your idea concerning this "limit" is interesting.
      In the message 17837 (about polynomials of the form (m.x - k)^2 + (m.x
      - k) + p ) i wrote this : " My contribution was not concerning
      quadratic polynomials givning less than p-1 distinct primes. An open
      problem is to find out polynomials of degree 2 giving L distinct
      primes when p > 41 and 40 < L < p-1 ".
      But what you are saying here, about polynomials of the form ax^2 - bx
      + c, is more general. You consider that the "limitation" of the lucky
      numbers of Euler will not concern the other quadratic polynomials (of
      the form ax^2 -bx + c) with more than 40 (i.e., the highest lucky
      number of Euler - 1) distinct positive prime numbers.
      So, by transforming my sentence above and using the polynomials ax^2 +
      bx+ c (to avoid any risk of confusion with ax^2 - bx + c), we could
      simply say : " an open problem is to find out polynomials of degree 2
      giving L distinct primes, with L > 40 ". In a first approach, this
      situation seems to be more general than mine : starting with your
      interpretation and the polynomials ax^2 + bx + c, with c = p prime, p
      > 41, L > 40 and x = 0 ... L-1, we have three possibilities :
      A. L < p-1
      B. L = p-1
      C. L > p-1
      The case A corresponds to my description above.
      The case B is impossible because p will never be a lucky number of
      Euler when p > 41.
      The case C paradoxically brings us to the case B : if a polynomial
      ax^2 + bx + p, with p > 41, gives more than p-1 distinct prime
      numbers, it gives already distinct prime numbers when we only consider
      x = 0 ... p-2.
      The interest of your proposal, Jens, is finally to confirm that the
      only possible polynomials to be sought are the polynomials included in
      A, where ax^2 + bx + p gives L distinct positive primes when p > 41
      and 40 < L < p-1.

      3. As i wrote above, the polynomials x^2 - x + c are often sources of
      problems. But the differences between x^2 - x + c and x^2 + x + c
      (when c is a lucky number of Euler) are not summarized with the fact
      that they are both giving p-1 distinct positive primes in a different
      range of the variable x (x = 1 ... c-1 for the first family
      and x = 0 ... c-2 for the second family).
      Something else appears when we compare them ...
      I would like to do a general remark on the nature of the independent
      term c when we look at general polynomials ax^2 + bx + c (with integer
      coefficients) giving any number of distinct positive prime numbers.
      We know that if we want a sequence of distinct and positive prime
      numbers, such that this sequence begins with x = 0, then we require
      that c is positive and prime. If it is not the case, then we have to
      change our way of presentation and begin all the sequences with x = 1
      (which is more logical in the sense that the FIRST value of x gives
      the FIRST value of the sequence). By the only fact that we are
      starting with x = 0, we artificially create a constraint on the nature
      of c, which must be prime in any case (i mean in all the polynomials
      giving any number of distinct positive primes). We do not have
      necessarily such a constraint when we decide that all the sequences
      must be counted from x = 1, and in this case we could imagine that c
      is not systematically a prime number.
      For instance, when i discussed about the polynomials (m.x - k)^2 +
      (m.x - k) + c, with m > 0 and k = 0, c-1, c-2, 2c-3 (see
      http://groups.yahoo.com/group/primenumbers/message/17841 ), i was
      obliged to consider that c is prime in each case of production of
      sequences of primes. Only because we require to start with w = 0. So
      all the families of polynomials giving p-1 prime numbers present
      consequently a prime independent term (a lucky number of Euler or
      'derivated' from it). Hence, the search for polynomials giving more
      than 40 positive distinct numbers could be strongly influenced by the
      initial conditions of counting.
      I don't know if this situation needs to do an adaptation of the
      definition of the prime-generating polynomials (perhaps the best is to
      continue not to precise how we are counting) or to be more interested
      in polynomials ax^2 - bx + c with an only aim of widening the
      possibilities for c (anyway i am not sure that they will give more
      good polynomials than ax^2 + bx +c).
      The attitude depends finally on what we are searching for : sequences
      with repeated prime numbers or ... or sequences with distinct positive
      prime numbers. For this last kind of search for instance, the
      polynomials x^2 - x + c , with c prime, are excluded when we count
      from x = 0 because they give a repeated prime number in the begin of
      the sequence.
      In conclusion, i would say that by starting the sequences at x = 1, it
      is not excluded that we enlarge the field of possibilities and give
      more chance to find out quadratic polynomials giving more than 40
      positive distinct prime numbers.

      Patrick Capelle.
    • Patrick Capelle
      ... The zoo of prime numbers did not accustom us to so much simplicity and absence of special constraints. In spite of similar formal aspects, we are far here
      Message 2 of 14 , Apr 1, 2006
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        --- In primenumbers@yahoogroups.com, Wed March 29, 2006, "Jens Kruse
        Andersen" <jens.k.a@...> wrote:
        > Heuristics support this plausible conjecture:
        > For any n>0 and k, there is a polynomial of degree n which
        > gives distinct positive primes for the first k integers.
        -------------------------------------------------------------------

        The zoo of prime numbers did not accustom us to so much simplicity and
        absence of special constraints.

        In spite of similar formal aspects, we are far here from the idea that
        for any k, there is a polynomial of degree n which gives distinct
        positive primes for the first k integers.

        Paradoxical situation : we have a lot of difficulties to find out one
        quadratic polynomial giving more than 40 distinct positive prime
        numbers , and during this time one can imagine the existence of a
        quadratic polynomial P(x) giving distinct and positive prime numbers
        when x goes from 0 to 10^10 ...

        Regards,
        Patrick Capelle.
      • gordon_as_number
        Only a comment. If you want to prove a polynomials P(x) is prime, isnt enough to determine the assoiated roots. E.g. P(x)=x^2+2 The number N is P(x) which can
        Message 3 of 14 , Apr 2, 2006
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          Only a comment.

          If you want to prove a polynomials P(x) is prime,
          isnt enough to determine the assoiated roots.

          E.g. P(x)=x^2+2

          The number N is P(x) which can base expanded in
          some base b as

          N=\sum_{i=0}^m c_i b^i .

          Determining the roots and the coefficients I claim
          is order \ln^3(N), which is a rough estimate here
          because I didnt let N vary and I didnt include what
          the value of m is. So you have to polynomials evaluated
          in the different bases b with the same coefficients c_i.
          The number of real c.c. pairs (x-\alpha_i)*(x-\alpha_j)^\star
          = integer or (x-\alpha_i) an integer span the prime factors.
          Arent there theorems that tell you when polynomials have
          non-integer root pairs so that you can make a statement
          towards their construction. I think there are. There
          should be large classes of integers N of the same degree
          and with the same coefficients c_j but with varying base;
          I think these can be constructed in ln^3 N_{max} time with
          N_{max} being the largest integer considered.

          The construction I refer to is given in Fast Factoring
          by Dr. Gordon Chalmers.
        • Phil Carmody
          ... First you need to specify the ring in which you re working. You do not do so. One cannot assume Z[x] given that many of the examples we ve seen have been
          Message 4 of 14 , Apr 2, 2006
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            --- gordon_as_number <gordon_as_number@...> wrote:
            > Only a comment.
            >
            > If you want to prove a polynomials P(x) is prime,

            First you need to specify the ring in which you're working.
            You do not do so.
            One cannot assume Z[x] given that many of the examples we've seen have
            been instead in Q[x].

            > isnt enough to determine the assoiated roots.
            >
            > E.g. P(x)=x^2+2
            >
            > The number N is P(x) which can base expanded in
            > some base b as
            >
            > N=\sum_{i=0}^m c_i b^i .

            Bases ought to be irrelevant. If they're not, you're doing something wrong.

            > Determining the roots and the coefficients I claim
            > is order \ln^3(N), which is a rough estimate here
            > because I didnt let N vary

            Order(funtion(N)) only makes sense if you let N vary.
            You've reached that level of illucidity again...

            > and I didnt include what
            > the value of m is. So you have to polynomials evaluated
            > in the different bases b

            Yup - illucid. Polynomials can be evaluated, but their value is
            independent of any base.

            > with the same coefficients c_i.

            The c_i, as you introduced them, are not coefficients.

            > The number of real c.c. pairs (x-\alpha_i)*(x-\alpha_j)^\star
            > = integer or (x-\alpha_i) an integer span the prime factors.

            With no firm foundation to build on, you've completely lost me now.
            No point in continuing.

            Phil

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